Wednesday 2 September 2015

Does the squishy affect the stretchy?

For my Ph.D. I wanted to compare my experimental measurements to a theoretical model. Unfortunately, the model I needed did not yet exist. This is the story of the collaboration that brought that new model into existence.

My graduate years were spent looking at DNA molecules squished into funny shaped tubes. They feature a very narrow slit, narrow enough to make the molecule spread out in two dimensions, and are filled with little pits, that the molecule can fall into to increase entropy. When a molecule falls into a certain number of pits, it looks like little tetris pieces. My master's was spent studying how DNA behaves in systems like this. My Ph.D. was spent using these systems to learn more about polymer physics.

Let the cavalcade of gifs begin! Each little bright square is a little less than a micron.

It's a big molecule but a small cavalcade.
DNA is a polymer. DNA is the best polymer.

A polymer is basically a molecular chain or string, with a path that curves around randomly. Polymers are generally in a reservoir with some temperature (in my case, in room temperature water), so they change orientations very quickly and randomly as water molecules bump into different parts of the chain. Entropy is effectively the number of ways a system can be rearranged and still look the same, and in the case of a polymer it's the number of ways you can reorient the path of the chain and still have the whole thing stay the same size. The maximum entropy configuration is what is called a "random coil." I would call it a blob, but that refers to something else.

When I study DNA stretched between these pits in a narrow slit, there are a few important things I have to take into account:

DNA is Stretchy

If you take the two ends of a polymer chain and hold them fixed, there are a finite number of ways to reorient the chain and still have the ends the same distance apart. If you pull them farther apart, there are fewer ways: you have decreased the entropy. Because the temperature of the system makes it constantly re-orient itself**, the ends will move closer together and entropy will increase. This is called an entropic force. This video paints a pretty good picture of initially stretched chains recoiling to their higher-entropy random coil configuration. Here is a gif from one of the first papers to visualize DNA doing it.

One of the first visualizations of stretched DNA relaxing to increase entropy, from Science vol. 264, 6 May 1994.

DNA is Squishy

So that's what happens when you stretch it. What happens when you squeeze it? The random coil has some preferred size, and when you put it in a tube smaller than that size, it has to squish itself in, and this again prevents certain configurations from forming, again lowering the entropy. This video (unfavourable format warning) shows this phenomenon: first DNA is driven into a narrow region where it is forced to adopt a low-entropy state, and then it recoils from the narrow region back into the reservoir to increase its entropy. Below is another nice video (that I did not make) of DNA being stretched from its random coil configuration then getting squished into a tube. To put it in human terms, each of our cells contains about 4 meters of coiled DNA. We have 100 trillion cells. There is enough DNA inside us to stretch to the sun 2000 times.

The Mystery of the Squishy Stretch

The question I wanted to know the answer to was: if DNA is squished into a tube, does that affect how hard it is to stretch? A general reason this might be the case is that the entropic force arises to resist the loss of entropy, but if entropy is already reduced by confinement, this restoring force might be less. In addition to genuine curiosity, I also needed a good model of stretched confined DNA to fully understand my measurements, which had some curiosities that I thought might be related to the stretchy-squishy conundrum.

When I was at the APS March Meeting, the largest yearly physics conference, in 2012 in Boston, I saw a talk from an Ottawa physicist Martin Bertrand about confined polymer simulations. I spoke to him, and told him about my experimental system, and asked if he was interested in simulating it, basically to verify the model I was using (I was still in a master's mode of thought). He then introduced me to Hendrick de Haan, also at Ottawa (now at Oshawa), who was actually interested in working with me to do simulations. I have been collaborating with him since. Initially we spoke just about doing some simulations of a molecule hopping between pits, but around this time my spring woes started to develop, so I asked him if he could figure out the problem I've described in the preceding paragraph.

I liked the idea of this project, because it would lead to a new idea that was interesting independently of my experiments. If Hendrick devoted his time and energy to simulating molecules hopping between pits {which he is more than happy to do :) }, and my experiments ended up going nowhere, the world would not benefit from any new ideas. If he figured out the stretchy-squishy theory, the world would gain knowledge whether my experiments worked or not.

It took a few years (it was not the main priority for either of us), but he figured it out, with colleagues Tyler Shendruk and David Sean. I will give a brief rundown of the theoretical insights that lead to this. There are two complementary methods to theoretical physics: pencil and paper derivations, which you use to get dandy expressions and equations for your theory, and numerical simulation, which you use to see how the big picture emerges from the fine details. Hendrick and friends used a combination of both, and the two techniques guided each other well.

Simulations are good for giving you the right answer if you ask the right question. Hendrick's molecular dynamics simulations basically simulated a chain of beads connected by springs with a certain bending rigidity. The beads receive random agitation from the "temperature" of the system, and everything moves around according to the thermal agitation and the forces acting on it. He simulated this for a range of heights and a range of forces, enough to get data of the force-extension relationship for all relevant scenarios. These serve as computational "data" against which a pencil-and-paper theory can be tested (real data is also nice, but experiments are not quite at the point of directly measuring this).

A stretched chain in a slit. If it looks rough it's because I copied a .mov from Hendrick's powerpoint slide then used a website to convert it to .gif and then used a different website to make the file smaller. Thanks Hendrick!

So how do we*** write down a theory that describes the squishy's effect on the stretchy? The Marko-Siggia equation gives the relationship between the force exerted on a polymer and the distance between the ends. In the limit of low forces, it behaves like a harmonic spring, and in the limit of high forces it behaves like a fluctuating string.

The Marko-Siggia formula. F is force, p is persistence length (50 nm for DNA), x is how far apart the ends are, Lc is the total length of the chain. If the units don't make sense, divide F by kT.

This equation describes how a stretched chain behaves in three dimensions, and someone had also derived how it behaves in two dimensions. Tyler and Hendrick focused on what happens between two and three dimensions, writing down an expression that simplifies to the 2D or 3D Marko-Siggia equations in the correct limit. Obviously we do not live in a fractional-dimension'ed universe, so what is the meaning of this fractional dimension?

To summarize, a polymer chain is half as floppy in two dimensions as in three, because there are less ways for its path to fluctuate. As the walls of a slit start to dictate the motion of a chain, they make it effectively stiffer as the chain is less likely to veer off its current path. So, this effective dimensionality is essentially a measure of how much less floppy the chain becomes as it is confined.

This is where the numerical simulations help guide the theory. By measuring the effective stiffness of an unstretched chain and various levels of confinement, they found a relationship for the effective dimensionality as a function of slit height. My experiments take place in slits between 50 and 200 nanometers, which corresponds to roughly 2.24 and 2.78.

The modified Marko-Siggia equation, interpolating between the 2D and 3D limits in a slit. d is effective dimensionality and h is slit height. This is actually the low force limit, read the paper to learn more!

So finally, there is an expression for a modified Marko-Siggia formula that describes a confined stretched chain. Does it work? The simplest thing to do is compare it to the numerical simulations, and we see that it does.

The simulation data matches the theory, and interpolates well between the 2D and 3D limits. Image taken from MacroLetters paper.

There is an additional subtlety however. If a chain is strongly stretched, its behaviour doesn't change that much when it is confined in a slit (imagine a chain stretched so much that it would never bounce into one of the would be like the walls aren't even there). To describe the stretchy-squishy interaction under high forces, they looked at it in terms of how the walls restrict certain Fourier modes of a vibrating string, and found a way of characterizing the effective dimensionality in terms of both height and stretch.

The paper was published on arXiv and then in the journal MacroLetters, which is inexplicably unavailable to Canadians. I am grateful that the editors and referees convinced the authors to switch from theorist units to actual units. I started using their version of the Marko-Siggia equation as part of my data analysis, and ultimately got some nice results that we published in a paper together. Even though I was not directly involved in the stretchy-squishy paper, I am proud to have been part of the process that lead to its creation.

I suppose I didn't actually answer the question in the title: yes, squishing DNA makes it easier to stretch.

**I don't feel the need to be consistent with my hyphenation.

***Here we refers to the writer guiding the readers through a process.

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